M-ary Signal Sets

The following signal set is used to transmit equally likely messages over an additive white Gaussian noise channel with spectral height $\frac{N_0}{2}$,

$$s_{i,j} = \sqrt{\frac{2E}{T}}(i \cdot \cos(2\pi f_c t) + j \... ...in(2\pi f_c t)) \;\;\mbox{for $0 \leq t \leq T$, $i,j=-1,0,1$.}$$

Thus, this signal set consists of M=9 signals.

1.

Draw and accurately label the signal constellation in an appropriately chosen signal space and indicate the decision boundaries formed by the optimum receiver.

2.

Compute the probability of error achieved by the optimum receiver.

3.

Assume that the energy of the transmitted signal can never exceed 2E. Is it possible to modify the above signal set in such a way that the probability of error is reduced without exceeding the limit on the signal energy? Explain why or why not.